MPRA Munich Personal RePEc Archive A Reference Dependent Representation with ubjective Tastes Andrei Barbos University of outh Florida 14. March 2008 Online at http://mpra.ub.uni-muenchen.de/23850/ MPRA Paper No. 23850, posted 13. July 2010 09:06 UTC
A Reference Dependent Representation with ubjective Tastes Andrei Barbos Department of Economics, University of outh Florida, Tampa, FL. February 2, 2010 Abstract Experimental and empirical evidence documents instances where the presence of an inferior option in a menu increases the attractiveness of the better options from that menu and thus distorts the normative ranking across menus. We analyze the case when besides this so called context effects bias there is also a concern for flexibility in the spirit of the literature initiated by Kreps (1979) and Dekel, Lipman and Rustichini (2001). ince the context effects bias and the desire for flexibility both increase the inclination of a decision maker to choose larger menus, the analysis allows the disentangling of the effect of the behavioral bias from the effect of the rational desire from flexibility. We find a weak condition on the set of ex post preferences under which the two effects are identifiable. We show that our representation is essentially unique. JEL Clasiffi cation: D11 Keywords: Context Effects; Reference Point Bias; ubjective Uncertainty. I would like to thank Marciano iniscalchi for very helpful comments and advice. I also thank comments from Eddie Dekel, Jeff Ely, Manuel Muller-Frank, Alessandro Pavan, Todd arver, Itai her, Asher Wolinsky and participants at the Bag Lunch Theory eminar at Northwestern University. All errors are mine. E-mail: a-barbos@northwestern.edu; Phone: (813)-974-6514; Fax: ( 813)-974-6510; Website: http://sites.google.com/site/andreibarbos/
1 Introduction Numerous observations from the marketing and psychology literature document the existence of a so called context-effects bias as suggested by the following experiment presented in imonson and Tversky (1992). The participants in the experiment were asked to choose between two substitute products, tissues and towels. Two versions of the experiment were designed. One superior brand of towels and one superior brand of tissues were included in both versions. In addition, in one version of the experiment the participants were offered with one inferior brand of towels, while in the other they were offered with one inferior brand of tissues. The results of the experiment showed that the market share of the superior quality brand was significantly higher when the inferior quality brand belonged to the same category. This example shows how the presence of an inferior option in a menu may make the better bundles of that menu appear more attractive by comparison, and thus distort the normative ranking of the available options. 1 This is a pattern of behavior inconsistent with the standard model of rationality, which posits that products which are never chosen for consumption should not influence the decision maker s choices. In a recent paper, Barbos (2009) studies a model of choice from categories, or menus, consistent with the above experimental evidence. ince menus are the objects of choice that reveal an individual s desire for flexibility, as a natural extension of the certainty model whose axiomatic foundations are provided in that paper, we study here the case when besides the context-effects bias, there is also a concern for flexibility in the spirit of the literature initiated by Kreps (1979) and Dekel, Lipman and Rustichini (2001) (henceforth DLR(2001)). More precisely, we analyze the case in which we allow for the presence of some underlying uncertainty between the moment of the choice of the menu and the moment of the choice of a specific option from within the menu. Allowing for uncertainty between the two stages makes sense especially in those applications in which there is some cost of switching between menus and the choice of the specific element from the menu is made either significantly later or repeatedly over a long period of time. In these cases, when choosing the menu the decision maker has to contemplate various potential realizations of his future preferences and thus, the usual intuition behind the notion of subjective tastes (Kreps (1979), DLR (2001)) applies here as well. 2 ince both the context effects bias and the desire for flexibility increase the inclination of a decision maker to choose a particular menu when that menu is expanded, the analysis in this paper allows 1 There are numerous similar observations in the psychology and experimental literature. ee for instance Bhargava, Kim and rivastava (2000), Hsee and Lecrerc (1998), Huber, Payne and Puto (1982) or Pan, O Curry and Pitts (1995). 2 Alternatively, one can think of this model as a study of the behavioral implications of a reference point bias in the model of subjective uncertainty introduced by DLR (2001). 2
disentangling the effect of the behavioral bias from the effect of the rational desire for flexibility in evaluating preferences over menus. We show that our representation is essentially unique. As a motivating example, consider an individual choosing between various movie rental services such as, Netflix, Vongo or Comcast On Demand. ome of these services offer a variety of plans, combining benefits and prices. There is no cost of switching between these plans, but there is a cost of switching between services. Thus, for instance Netflix requires an investment in a DVD player, Vongo requires a broadband internet connection and Comcast On Demand requires a subscription to the cable service. Assume that after some careful consideration, a consumer decides to subscribe to Netflix and then selects one of the available options regarding the number of DVDs he can rent at a time. Now, it may happen that the consumer is uncertain about which of the various options fits him best and needs to test some of them. In this case, since there is no cost of switching between the various plans offered by each service, it makes sense to regard the initial selection of rental service as a choice between menus, and the selection of a plan as a choice from a menu made after some uncertainty is resolved. Thus introducing uncertainty in a model that attempts to capture the agent s behavior is necessary. For this model, following Kreps (1979), we will identify a menu with an ex ante observable action that after some subjective uncertainty is resolved will make a certain set of outcomes available ex post. The observability of these ex ante actions renders the preference over menus a revealed preference; thus, we can take this to be our primitive in the uncertainty setting. The reference dependent representation under uncertainty that we will axiomatize is the following: V (A) = [ ] [ ] maxu(z, s) µ(ds) θmin U(x, s)µ(ds) z A x A where is a state space capturing the subjective uncertainty with µ a positive measure over, U(z, s) is the ex post state utility of option z in state s and θ is a parameter that measures the strength of the behavioral bias. 3 The space will satisfy an additional condition that will specify that the decision maker does not reverse or almost reverse his ex ante preferences. Thus, while we allow for the presence of uncertainty, we do restrict attention to those applications in which there exists some underlying phenomenon that makes the ex ante preferences relevant for the ex post stage. The behavioral bias is identifiable only in those applications in which this condition is satisfied. We study the behavioral implications of both a finite and an infinite state space. An infinite state space appears, for instance, in models in which the individual has a continuous distribution of the ex post tastes over the characteristics of the available options. (1) 3 Note that the reference point is the ex ante least preferred option from within the menu. 3
Now, it is straightforward to see that (1) can be written as: [ V (A) = (1 θ) maxu(z, s) z A ] [ µ(ds) + θ max z A U(z, s) min x A ] U(x, s) µ(ds) (2) Thus, the preference for a menu is determined by the combination of a normative component and a behavioral bias component. The normative component is the weighted average of the utilities of the normative best options from the menu in each of the possible ex post states. The weighting factors are the subjective probabilities of these ex post states. The behavioral bias component is the weighted average of the difference in utilities between the normatively ex post best options in the menu and the normatively ex ante worst element in the menu. The second component is a measure of the increase in the relative attractiveness of the better options from a menu generated by the presence of the inferior option against which they are compared. We show that the main axiom that captures the departure from rationality in the certainty model from Barbos(2009) is almost suffi cient to deliver the context effects representation under uncertainty when added to the standard axioms from DLR (2001). More precisely, the axiom is suffi cient for the case of a finite state space. When the state space is infinite, for the behavioral bias to be identifiable, an additional simple axiom is required. This additional axiom imposes the existence of a pair consisting of a menu A and a lottery y such that y does not provide any ex post flexibility to a decision maker that was faced initially with the menu A.This axiom is equivalent to the condition that the decision maker does not reverse or almost reverse his ex ante preferences. Under this assumption, the effect of the behavioral bias can then be measured by studying the effect of expanding the menu A with the lottery y. As the ex post preferences are not observable, the existence of this pair is imposed through some ex ante behavioral implications. 4 Preferences over menus were considered for the first time by Kreps (1979). He identified an act with the choice of a set of future options out of which at a later stage the decision maker chooses his most preferred element. He interpreted the agent s preference for the flexibility offered by the menu as being generated by some underlying subjective uncertainty that will be resolved between the moment when the choice of the menu is made and the moment when the choice from the menu is made. This allowed him to show that under suffi ciently weak conditions, the decision maker behaves as if the uncertainty were described by a subjective state space, where each state is identified with an ex post subjective utility. The preferences that we study in this paper belong to the class of preferences modeled by DLR (2001). DLR (2001) considered menus of lotteries instead of menus of deterministic bundles; this allowed 4 When the state space is finite, we show that a pair (A, y) with the required property always exists and thus this additional axiom is not necessary. 4
restricting the ex post state utilities to be of the expected utility form. This addressed the issue of the nonuniqueness of the subjective state space characteristic to the Kreps representation. Also, unlike Kreps(1979), DLR (2001) allowed for subjective states of negative measure to capture not only a preference for flexibility but also a preference for commitment. There is a large body of literature that built on the class of preferences introduced by DLR (2001). Gul and Pesendorfer(2001) were the first to give meaning to the abstract subjective states derived in earlier papers. Thus, they imposed conditions on preferences such that the resulting state space consists of one state of negative measure representing a temptation preference and one state of positive measure representing the second period preferences which combine a normative preference and the temptation preference. This combination of normative and temptation preferences has been implemented in the meantime in other papers to model various behavioral biases, such as non-bayesian updating, cognitive dissonance, etc. For examples, see Epstein and Kopylov (2007) or Kopylov and Noor (2009). In another direction, the preference for commitment has been interpreted in arver(2008) not as being driven by the presence of temptation but by the anticipation of regret. The rest of the paper is organized as follows. In section 2 we present the basic assumptions common to most of the literature on preferences over menus. Also, in ection 2 we define the representation in our model and compare it with other representations that built on the DLR (2001) framework. In section 3 we present our additional axioms and the main results which state the equivalence between the axioms and the representations, while section 4 concludes the paper. Most proofs are relegated to the Appendix. 2 The Framework and the Representations Let Z be a finite space of outcomes or prizes and let (Z) denote the set of probability measures on Z endowed with the topology of convergence in distribution. Let K( (Z)) denote the collection of all nonempty closed subsets of (Z). Endowing K( (Z)) with the Hausdorff topology we make it a compact metric space. Elements of (Z) will be called lotteries and will be denoted by x, y, z, etc., while the typical elements K( (Z)) will be called menus and will be denoted by A, B, C, etc. The decision maker is assumed to have a revealed preference relation over the elements in K( (Z)). For any two menus A, B K( (Z)) and any α [0, 1], define their convex combination as αa + (1 α)b {z (Z) : z = αx + (1 α)y, for some x A and y B}. We will impose throughout the paper the following standard axioms on the preference. For a detailed interpretation and motivation of these axioms, see DLR(2001). 5
Axiom 1 (Weak Order). is a complete and transitive binary relation. Axiom 2 (Continuity). For any A K( (Z)), the upper and lower contour sets, {B K( (Z)) : B A} and {B K( (Z)) : B A}, are closed in the Hausdorff metric topology. Axiom 3 (Independence). αa + (1 α)c αb + (1 α)c. For all A, B, C K( (Z)) and any α (0, 1), A B implies The representation in (1) is a particular case of the additive expected utility representation as defined and axiomatized for the first time by DLR (2001). We will also frequently refer to it throughout as the DLR representation. Definition 4 An additive expected utility representation of is a nonempty possibly infinite set, a state dependent utility function U : (Z) R and a finitely additive signed Borel measure µ on, such that V : K( (Z)) R, defined for all A K( (Z)) by V (A) = [ ] maxu(z, s) µ(ds) (3) z A is continuous and represents and each U(, s) is an expected utility function in that for each s there exists u s : Z R such that U(z, s) = z u s. Following DLR (2001), we allow for the uncertainty to be completely subjective. Thus, besides allowing for a subjective distribution over the ex post contingencies as in standard avage type models, we also allow for the actual space of ex post contingencies to be subjective. ee DLR (2001) for further details. uniquely identified by the corresponding ex post state utility. as: Therefore, a state in the above representation can be For the case when the state space is finite the representation can be equivalently written V (A) = [ ] [ ] [ ] maxu(z, s) µ(s) = s + max u s(z) s max u s(z) s z A z A z A where + {s : µ(s) > 0} and {s : µ(s) < 0} and u s ( ) µ(s) U(, s). In writing the above we used the fact that the measure over states and the state utility are not separately identified in models of state-dependent utility, so they can be combined together. Note that the above definition allows the measure µ over the states to be signed. DLR (2001) call positive states and negative states, the states in the support of the positively signed and respectively negatively signed components of µ. Intuitively, as stated in DLR (2001), the (4) 6
positive states would reveal the agent s desire for flexibility, while the negative states would reveal his desire for commitment. In our paper, unlike the other papers building on the DLR (2001) framework, the agent is assumed to not have any kind of commitment issues. Therefore we assume throughout an additional axiom on preferences called Monotonicity, which imposes that weakly larger sets in the partial order given by inclusion be weakly preferred by the decision maker. 5 This is a condition consistent with the assumption of the agent not experiencing commitment problems. Axiom 5 (Monotonicity). For all A, B K( (Z)) with A B, we have B A. DLR (2001) and Dekel, Lipman, Rustichini and arver(2007) prove the following result. Theorem 6 When the set Z is finite, the preference has an additive expected utility representation with a measure µ which is always positive if and only if it satisfies Weak Order, Independence, Continuity, and Monotonicity. The effects of imposing Monotonicity on preferences are the following. Firstly, the axiom insures that the measure over the states from the representation is everywhere positive. econdly, it allows us to obtain a stronger property of the measure, that is σ-additivity instead of finite additivity as in DLR (2001). Finally, Dekel, Lipman, Rustichini and arver (2007) show that if Monotonicity is not imposed, the Continuity axiom as presented above needs to be strenghtened to an axiom which they call trong Continuity in order to get the additive expected utility representation with a signed measure. The additional condition on preferences that is needed delivers the Lipschitz continuity of the representation. Here, since we do assume Monotonicity, we may impose the weaker continuity condition given by the Continuity axiom presented above. We will also consider the case when the state space from the DLR representation is finite. A necessary and suffi cient condition to obtain a finite state space was found in Dekel, Lipman and Rustichini (2009). The authors call this additional axiom Finiteness. 6 Axiom 7 (Finiteness) Every menu A K( (Z)) has a finite critical set, where a critical set of a menu A is a any set A such that for all B with A hull(b) hull(a) we have B A. 5 The Monotonicity axiom is part of the axiomatization of the preference for flexibility in Kreps (1979). 6 hull(a) = {z Z : z = k i=1 λ iz i with λ i 0, k i=1 λ i = 1 and z i A} denotes the convex hull of a set A. 7
Dekel, Lipman and Rustichini (2009) prove the following result. Theorem 8 When the set Z is finite, the preference has an additive expected utility representation with a measure µ which is always positive and with a finite state space if and only if it satisfies Weak Order, Independence, Continuity, Monotonicity and Finiteness. The Reference Dependent Representation under Uncertainty The specifics of the representation from DLR (2001) require a number of normalizations. First, as explained in DLR (2001) the state space is just an index set that allows reference to different ex post preferences. Moreover, the ex post state utilities which are assumed to be of the expected utility form are identified only up to affi ne transformations, so we follow the approach in DLR (2001) and restrict the state space to the set of normalized utilities N {s R N : N k=1 sk = 0 and N } k=1 (sk ) 2 = 1. (5) Throughout the rest of the paper we use s N to refer both to a second period contingency as well as to the normalized expected utility function representing the preferences in that state. Thus, the utility of x (Z) in state s will be U(x, s) = x s = N k=1 xk s k, where s = (s 1,..., s N ) N is the normalized expected utility function that represents the state s preferences. Note now that the restrictions of the Weak Order, Continuity and Independence axioms to (Z) imply by standard results the existence of an expected utility function v( ) that represents the restriction of to (Z). arver(2008) shows that since N contains the normalization of any affi ne function on (Z), there exists s N and λ 0 such that v(x) = λ N k=1 xk s k for all x (Z). We define now formally a normalized representation of the preferences exhibiting the reference point bias. As mentioned in the Introduction, we assume that in the second period, after the uncertainty is resolved the decision maker cannot reverse or almost reverse his ex ante tastes. Denote the ball of radius ε around s where ε > 0 and s by N ε (s) {s : d(s, s) < ε} where d(, ) is the usual Euclidean metric in R N. Definition 9 Let Z be any finite set. A normalized reference-dependent representation under uncertainty of consists of a nonempty possibly infinite measurable set N, a Borel measure µ on N, with being the unique support 7 of µ and a constant θ (0, 1), such that 7 The support of a Borel σ-additive measure µ, if it exists, is a closed set, denoted, satisfying: (1) µ( c ) = 0; 8
(i) V : K( (Z)) R, defined for all A K( (Z)) by V (A) = [ ] [ ] max(z s) µ(ds) θmin (x s)µ(ds) z A x A (6) represents the preference ; (ii) the utility of a lottery x (Z) in state s is x s; (iii) if s N is the normalized utility that represents the restriction of to (Z) then there exists ε > 0 such that N \N ε ( s ). 8 We emphasize that besides and µ which are the usual elements in a normalized DLR representation, the parameter θ and the restriction (iii) on the set of ex post utility functions will also be deduced from preference as a part of the representation. Note that the functional form in (6) for V ( ) can be rewritten as: V (A) = [ ] [ ] max(z s) µ(ds) + max θ (x s)µ(ds), (7) z A x A and thus our representation is indeed a particular form of an additive expected utility representation with all states having associated a positive measure. The condition (iii) on the set of ex post utilities allows identification of the behavioral bias modeled by our representation with the reference point bias. Note that (6) implies that the ex ante preferences over singletons are represented by the utility function v(x) (1 θ) (x s)µ(ds). By inspecting (7), it is clear that the preferences represented by v (x) (x s)µ(ds) could constitute just another ex post state in a DLR(2001) framework with the property that these ex post preferences are exactly the reverse of the ex ante preferences over singletons. We rule out this possibility by making the arguably reasonable assumption that in the second period, after the uncertainty is resolved, the decision maker cannot reverse [ or almost reverse ] his ex ante tastes. This is done by the identification of the term max θ (x s)µ(ds) from the equivalent representation in x A (7) with the impact of a reference-point bias and by the condition (iii) from Definition 9. In the remaining of this ection we present the particular structure imposed on the ex post states of the DLR representation by various papers that built on that framework to underline and (2) If G is open and G, then µ(g ) > 0. Theorem 10.13 in Aliprantis and Border (1999) shows that if the underlying topological space on which µ is defined is second countable or if µ is tight, then µ has a (unique) support. In our case, N is clearly second countable, so the definition is correct. 8 When the state space is finite, condition (iii) can be written as s /. Also, note that s N. 9
the differences between these representations and ours. 9 Note firstly from (7) that in our model there exist no negative states and there exists one positive state of strictly positive measure having the corresponding utility a negative affi ne transformation of the utilities of the rest of the states. In Gul and Pesendorfer (2001) the equivalent DLR representation is the following: V (A) = max x A u 1(x) max y A u 2(y) (8) where u 1 is the utility that represents the second period preference relation and u 2 is the temptation component of these second period preferences. Therefore, in this representation there is one positive state and one negative state with no particular mathematical relation between them. In arver (2008) the equivalent DLR representation of his regret representation is: [ ] [ ] V (A) = max (1 + K) U(z, s)µ(ds) maxku(x, s) µ(ds) (9) z A x A where K 0. Thus, in this case there is a number, possibly infinite of negative states and one positive state whose corresponding state utility is a positive affi ne transformation of the utilities corresponding to the negative states. 10 The equivalent representation from Dekel, Lipman and Rustichini (2009) of what they call the temptation representation is: V (A) = [ ] maxu(z, s) µ(s) { s z A s j J s [ ]} maxu(y, j) µ(s) (10) y A which is a generalization of the one from Gul and Pesendorfer (2001) in the sense that it assumes multiple ex post states and for each ex post state multiple ex post temptations. This representation has a number of positive states and for each positive state a number of corresponding negative states with some underlying structure among them. Unlike the other representations, in Dekel, Lipman and Rustichini (2009) the state space is assumed to be finite. 9 We emphasize that the actual representation from each of the papers is different from that presented here as being its equivalent DLR representation in order to capture the corresponding behavioral trait that is analyzed in each paper. 10 The regret representation as defined in arver (2008) has the Borel measure µ positive. However, as mentioned in that paper as well, the equivalent DLR representation is signed and has a negative component. 10
3 The Axioms and the Main Results As noted before, the representation in (6) is a special type of an additive expected utility representation with a positive measure. Thus, it will be necessary that the preference satisfy Weak Order, Continuity, Independence and Monotonicity. Two additional axioms will be suffi cient for the preference to have a reference-dependent representation when combined with the above four axioms. The first additional axiom captures the departure from the standard model of rationality that we study in this paper. Axiom 10 (CEB: Context-Eff ects Bias) : For any pair (A, x) K( (Z)) (Z), such that {y} {x} for all y A, we have A {x} A. Axiom CEB states that if the decision maker has the set of possible choices A expanded by adding an option, say a singleton {x}, which from an ex ante point of view is strictly worse that the rest of the elements in the menu, then the agent will strictly prefer the new expanded set A {x} to the initial one A. The motivation for this preference is given by the fact that the inferior lottery x will be chosen as the new reference point and thus the overall attractiveness of the better options from the menu will increase. Note that since the preferences in the text of the axiom are strict, Axiom CEB imposes that the agent has a strict preference for menus having additional strictly inferior outcomes. This corresponds to the restriction that θ > 0 in the representation in (6). Axiom CEB provides the departure from the standard model of rationality as suggested by the presence of a behavioral bias. However, when allowing for an infinite state space this departure is identified only when combined with the Axiom CEB-2 presented below. This is because, when allowing for the presence of uncertainty, it may happen that an ex ante inferior option still provides some ex post flexibility to the elements of a set, and thus the pattern of choice suggested by Axiom CEB is valid without assuming any reference point bias. In order to have a departure from the standard rational preferences, there must exist at least one set A and at least one lottery y strictly worse from an ex ante point of view to all elements of A such that in any ex post state there exists an element in A that is at least as preferred as y. Then, imposing Axiom CEB to the sets A and A {y} would provide the departure. Now, in the case of a finite state space, the pair (A, y) with the desired properties always exists when we maintain the assumption that the second period preferences cannot be exactly the reversed ex ante preferences. 11 In the case of an infinite state space, the existence of such a pair (A, y) will be imposed by Axiom CEB-2 below. As the ex post preferences are not observable, 11 ee the necessity part of the proof of Theorem 14 below for a formal argument. 11
this is done by imposing a natural implication of the existence of the pair (A, y) on the ex ante preferences. Before presenting Axiom CEB-2 we will make a remark that suggests that imposing axiomatically the existence of such a pair is correct when the preferences that we are studying are represented by a utility function as in (6). Remark 11 When the preferences admit a normalized reference-dependent representation as in (6), there exist a set A K( (Z)) with A int( (Z)) and a lottery y (Z) such that: (i) for any x A we have x s > y s and (ii) for any s there exists x A such that x s > y s. Proof. ee Appendix A1 for some notation on support functionals and then Lemma 29. The second non standard axiom for the case of an infinite subjective state space is the following. Axiom 12 (CEB-2): There exists a set A K( (Z)) with A int( (Z)) and a set B K( (Z)) with A B and {x} {y} for all x A and some y B, such that for all lotteries z (Z) with {y} {z} for all y B, we have B {z} A {z}. To see the motivation for this axiom, consider a pair {A, y} such that y does not provide any ex post flexibility to A. Then, on the one hand, by choice of y, it provides no ex post flexibility to the set A {z}. On the other, since y is weakly preferred to z from an ex ante point of view, y will not be the reference point chosen from A {y, z}. ince under no circumstances the agent would choose y over the elements in A {z}, he is as well off having at hand the menu A {z} as he is having the larger menu A {y, z}. Therefore, we impose the required indifference. The condition that B is some superset of A instead of A {y} is a nonessential weakening of the axiom meant only to simplify the notation in the proof of the main theorem when we characterize a menu by the corresponding support functionals. We mention that while the condition from the text of the axiom is valid for an infinite number of sets in K( (Z)), we do not impose this condition to hold for all these sets simply because there may exist A K( (Z)) such that there exists no lottery y / A that does not provide any ex post flexibility to A. Also, we do not impose the indifference of the sets constructed as in the text of Axiom CEB-2, but for all B that contain a lottery y such that y is strictly less preferred to all elements of A from the ex ante point of view. This is because, as mentioned earlier, a lottery which is ex ante inferior to all x A could turn out ex post to be better to all elements of A and then B {z} A {z}. The weak restriction imposed in the text of the Axiom that the condition is valid for at least one pair (A, B) is suffi cient to obtain 12
the desired representation for all sets due to the additional structure provided by the EU form of the ex post utilities. Note also that we impose that an element of B be strictly worse than all elements of A. Without this condition, it is clear that the axiom would not have any bite since we could always let B be exactly the set A. Now we are ready to state the main result of the paper: Theorem 13 The preference has a representation as in (6) if and only if it satisfies Weak Order, Continuity, Independence, Monotonicity, Axiom CEB and Axiom CEB-2. Proof. ee Appendix A1 and Appendix A2. While the complete proof of Theorem 13 can be found in the Appendix, we present here for intuition a sketch of this proof. We start by showing how the argument goes if the state space were finite. This reveals how Axiom CEB and Axiom CEB-2 work to give us the representation. Weak Order, Continuity, Independence and Monotonicity imply that the preference over menus has the following representation, with µ a positive measure: V (A) = [ ] max (z s) µ(s), for all A K( (Z)) (11) s z A Denote by v( ) the restriction of V ( ) to singletons. Thus, v represents the ex ante preference over lotteries and by (11) we have: v(z) = (z s)µ(s), for all z (Z). Moreover, as s claimed in ection 2, since v is an affi ne function, there exists s N, such that v(z) = λ(z s ) for all z (Z). Now, recall that the representation in (6) is a particular case of a DLR representation in which the utility associated with one of the states is a negative affi ne transformation of the utilities associated with the rest of the states. We will prove here that under Axiom CEB and Axiom CEB-2, the representation in (11) must have exactly that structure on the ex post states, which comes down to showing that s. The rest of the proof consists of showing that given that structure, the representation in (11) can be written as in (6). This second part of the proof is just simple algebra manipulations and its presentation is relegated to the Appendix. Let A int(k( (Z))) and B K( (Z)) be as in the definition of Axiom CEB-2, that is A B and {x} {y} for all x A and some y B. Then by Axiom CEB and Monotonicity we have B A; using the representation in (11) it follows that: s [ ] max (x s) µ(s) > [ ] max (x s) µ(s). (12) x B s x A 13
ince the measure µ is positive, (12) implies that there must exist s such that max (x x B s ) > max (x x A s ). Denote the strict lower contour sets associated with an expected utility function s and a lottery y (Z) by L s (y) {x (Z) : x s < y s}. Then, given the linearity of the utility functions, a state utility s will be a negative affi ne transformation of s if and only if L s (y) L s (y) = for all y (Z). Assume by contradiction that there is no such state utility as the one that we are looking for, that is L s (y) L s (y) for all s. We show that in this case, if Axiom CEB holds then Axiom CEB-2 must be violated. Take some y B such that x s y s for all x B and then some z L s (y) L s (y) which is nonempty by the contradiction assumption. Then, since y B we will have max (x x B s ) > z s ( ) so max (x x B {z} s ) = max (x x B s ) > max max (x x A s ), z s = max (x x A {z} s ). Therefore: V (B {z}) = [ ] max (x s) µ(s) + µ(s ) max (x s \{s } x B {z} x B {z} s ) > s \{s } s \{s } [ [ ] max (x s) µ(s) + µ(s ) max (x x B {z} x A {z} s ) ] max (x s) x A {z} µ(s) + µ(s ) max x A {z} (x s ) = V (A {z}). Therefore, V (B {z}) > V (A {z}) so there exists z with x s > z s for all x B such that B {z} A {z} which violates Axiom CEB-2 as claimed. In conclusion, there must exist a state s that is a negative affi ne transformation of s. Proving the necessity of Axiom CEB is straightforward. To see that Axiom CEB-2 must also be satisfied when the preferences can be represented as in (6) with a finite state space, take some lottery y int( (Z)) and for each s, take x s H s (y) L s (y) (Z), where H s (y) {x (Z) : x s = y s}. Let A s x s and B A {y}. Then, by the choice of the set A, we will have {x} {y} for all x A. On the other hand, for any z L s (y) H s (y) we have V (A {z}) = [ ] ] max (w s) µ(ds) θ min s w A {z} w A {z} [ s (w s)µ(ds) = s [ max [ ( )] ] max max(w s), z s µ(ds) θ s w A [ s (z s)µ(ds) = )] ] max(w s), z s µ(ds) θ min w B [ s (w s)µ(ds) = V (B {z}). ( w B {z} 14
In the above we used the fact that the restriction of the representation to singletons implies {x} {y} (1 θ) (x s) > (1 θ) (x s) and the fact that since for each s, s s there exists x s A H s (y) we have (w s) = max(w s). max w A {y} While in the case of a finite state space, it is suffi cient to show that s must be one of the states from the DLR representation, for the case of an infinite state space this is not enough. This is because the state s can always be added to the state space and assign a measure zero. Thus, for the proof of the suffi ciency of the axioms for the infinite state space case, the main challenges are to show that the DLR measure of s is strictly positive and to show the existence of the empty neighborhood of s. The first straightforward step in the general proof is to use Axiom CEB to assert the existence of a set Ŝ1 with µ(ŝ1) > 0, such that max (x x B s ) > max (x x A s ) for all s Ŝ1 (see Lemma 18). The main goal of the rest of the proof of suffi ciency is to construct a set Ŝ5 Ŝ1 with µ(ŝ5) > 0 and find a lottery z (Z), such that max (x x B s ) > z s for all s Ŝ5 and min (x s ) > z s. Then, an argument similar to x B the one from the case of a finite state space will complete the proof by showing that Axiom CEB-2 must be violated. Now, note the following fact. Fact: If s {s1,...,s n}l s (y) and ( s {s1,...,s n}l s (y) ) L sn+1 (y) = for some y (Z), then s n+1 hull( s 1,..., s n ). Thus, if we could find a set of N 1 linearly independent ex post states in Ŝ1with: w A µ(int (hull({s 1,..., s N 1 }))) > 0 (13) we could then first argue inductively using the above Fact that: s {s1,...,s N 1 }L s (y) (14) (see Lemma 25) and then also argue that s hull({s1,...,s N 1 })L s (y) (see Lemma 26). In addition, if we ensure that: s / hull({ s 1,..., s N 1 }) (15) then again by the above Fact, we would also have that ( s {s1,...,s N 1 }L s (y) ) L s (y) and thus be able to take: z ( s hull({s1,...,s N 1 })L s (y) ) L s (y) (16) Finally, by letting: Ŝ 5 hull({s 1,..., s N 1 }) (17) 15
and selecting z as above, we would achieve the desired result. However, there are two problems with this approach that do not allow the argument to go through as stated. Firstly, since Axiom CEB-2 does not state the existence of a lottery y, but of a set B, the above mentioned Fact is not true if we replace y with the set B. econdly, since N is the subset of normalized utilities, finding N 1 linearly independent values with the desired properties is not immediately obvious, if possible at all. In order to solve the first issue, we will expand the set (Z) to the smallest affi ne set that contains it, that is to Ω {z R N : N i=1 z i = 1}. Then, by defining the expanded lower contour sets L s (B) {y Ω : y s < z s for all z B}, we will be able to prove a counterpart of the above Fact (see Lemma 24). While in the end, we will obtain an element z that belongs to the intersection of these expanded sets ( s Ŝ 5 L s (B) ) L s (B) (see Lemma 26), since (Z) has a non empty algebraic interior in Ω, we will be able to select a lottery z (Z) that will have the desired properties (see first Lemma 17 to see why we may consider without loss of generality that B int( (Z)) in the text of Axiom CEB-2 and then Lemma 27). 12 To solve the second problem, we will expand the set of ex post utilities from the normalized set N to the set P N = {s R N : N } k=1 sk = 0. A first effect of this expansion is that the counterpart of the above Fact will have now to be written in terms of convex cones instead of convex hulls (see Lemma 24). But since z ( s {s1,...,s N 1 }L s (B) ) L s (B) still implies z ( s cone({s1,...,s N 1 })L s (B) ) L s (B) the argument will continue to go through (see Lemma 26). Therefore, we need to find N 1 linearly independent utilities in P N such that µ(int (cone({s 1,..., s N 1 })) Ŝ1) > 0 and s / cone({ s 1,..., s N 1 }). Then, by letting Ŝ5 int (cone({s 1,..., s N 1 })) Ŝ1 and z be selected as explained above we would complete the argument. We mention here that considering N 1 linearly independent states that would include s would not solve the problem because then the interior of the cone generated by less than N 1 states in the N 1 dimensional space P N would be empty and thus of zero measure. Therefore, the need for the more elaborate construction. Now, using the contradiction assumption µ({ s }) = 0, we show in Lemma ) 19 and Lemma 20 that there exists a set Ŝ2 Ŝ1 such that µ(ŝ2) > 0 and s / cone (Ŝ2. Next, Lemma 21 and Lemma 22 show that there exists a set Ŝ3 Ŝ2 and ɛ > 0 such that s / cone( s Ŝ 3 N ɛ (s)), where N ɛ (s) is the closed ball of radius ɛ around s. In Lemma 23 we construct the set of linearly independent utilities {s 1,..., s N } with the properties presented above. This is done by covering the compact set Ŝ3 with a finite partition extracted from a cover of Ŝ3 whose elements are the intersections of this set Ŝ3 with the convex cones generated for each s Ŝ 3 by some set of N 1 linearly independent utilities {s s 1,..., s s N 1 } with the property that ( ) 12 Note that while z s Ŝ 5 L s (B) L s (B), z will satisfy only the weaker set of properties max (x x B s ) > z s for all s Ŝ5 and z L s (B). As argued above, this is suffi cient to obtain the desired result. 16
s int ( cone ( {s s 1,..., s s N 1 })). In addition, since s is suffi ciently far away from the set cone( s Ŝ 3 N ɛ (s)), as ensured in the Lemma 22, we will have s / cone({ s 1,..., s N 1 }). The proof of the existence of the empty neighborhood around s is presented in Theorem 28. It differs from the proof of the fact that µ({ s }) > 0 only in the way in which it uses the contradiction assumption in the proof of Lemma 19 as a first step in the construction of the set Ŝ 2. We will present in the Appendix only the part of the argument in which the two proofs are different. The necessity part of the proof of Theorem 13 also needs a rather elaborate approach. This is because the infinite set s x s, with x s chosen as in the intuitive argument presented above, is not necessarily closed and thus not necessarily compact. Thus, we need to take A = cl( s x s ). But then the fact that we select x s L s (y) for each s does not necessarily imply {x} {y} for all x A and this invalidates the required conclusion. Part (iii) of the Definition 9 will help overcome this problem but the construction is still not straightforward. The proof of the necessity will share some steps which are similar to steps from the proof of Theorem 16 and those steps will be presented without proof in the Appendix. However, we emphasize here that while in both proofs the initial steps consist of partitioning some compact set of ex post states into a finite number of subsets such that the states contained in each subset share some common properties, there is an important difference in terms of the ultimate goal of these arguments. Thus, in the proof of Theorem 16, which is the result that shows that µ({ s }) > 0, the partitioning of Ŝ1 was done so that we could in the end claim that one of these subsets, namely Ŝ5, must be of strictly positive measure since the measure of the set Ŝ1 was strictly positive. In the proof of the necessity of Axiom CEB-2, the goal is to partition the set N \N ε ( s ) into a finite number of subsets, each sharing some common relevant properties, so that we can resolve the problems raised by the infiniteness of the state space. We defer the presentation of the argument to the Appendix. Next, we present an additional result that constitutes the representation theorem for the case of a finite ex post state space of uncertainty. As argued above, in this case the restriction on preferences given by Axiom CEB-2 is not necessary. To prove the theorem below, it is enough to show that by replacing Axiom CEB-2 with Finiteness, the resulting set of axioms imply Axiom CEB-2. Then the argument from the sketch of the proof of Theorem 13 would complete the proof. howing that Axiom CEB-2 must be satisfied in this case can be done by following an argument close to the one used above as the intuitive proof of Theorem 13 for the finite case. Theorem 14 The preference has a representation as in (6) with a finite state space if and only if it satisfies Weak Order, Continuity, Independence, Monotonicity, Axiom CEB and Finiteness. 17
We close this section with a result that states the uniqueness of the representation for the uncertainty model. This result is important because it allows the interpretation the objects of the representation as intended. Thus, the fact that the parameters of the representation are identified ensures that when observing choice, it is feasible to disentangle the impact on behavior of the context effects from the impact of the presence of subjective uncertainty. Our representation in (6) is identified by the elements of the set (µ, θ) where µ is a probability over the N and θ measures the strength of the behavioral bias. The following theorem shows that both µ and θ are identified from preferences. Theorem 15 uppose that (µ 1, θ 1 ) is a normalized representation of some preferences satisfying Weak Order, Continuity, Independence, Monotonicity, Axiom CEB and Axiom CEB-2. Then, if (µ 2, θ 2 ) is also a normalized representation of we must have θ 1 = θ 2 and µ 1 = µ 2. Proof. ee Appendix A3. 4 Conclusion This paper studies a model of reference-dependent preferences over menus of lotteries. We extend the model of choices from categories in Barbos (2009) to allow for the presence of some underlying subjective uncertainty between the moment of the choice of a menu and the time a specific option within the menu is selected. The axiomatic foundations of this model allow for the disentangling of the context effects bias from the rational desire from flexibility that is usually captured by preferences over menus. We identify a weak condition on the set of ex post preferences under which the two effects are distinguishable from each other. We find the behavioral implications of both a finite and an infinite space of uncertainty. Appendix A1. Construction of the state space for the uncertainty model We present here briefly the construction of the state space from Dekel, Lipman and Rustichini (2001) as we will utilize the concepts introduced there extensively in the rest of the proof. Firstly, as shown in DLR (2001) under Weak Order, Continuity and Independence any set of lotteries in (Z) is indifferent to its convex hull. Thus, we can restrict attention to the set 18
of convex sets 13 in K( (Z)), which we denote from now on with K( (Z)). Recall that the number of outcomes in Z is denoted by N and that N is the set of normalized expected utility functions on (Z). Define by C( N ) the set of real-valued continuous functions on N endow it with the topology given by the sup-norm metric. Embed K( (Z)) into C( N ) by identifying each menu with its support function: A σ A, with σ A (s) = max N x A k=1 xk s k. It is a standard result that the above mapping is an embedding, one-to-one and monotonic. Thus, for all A, B K( (Z)), σ A ( ) = σ B ( ) implies A = B and A B implies σ A σ B. The order used on C( N ) is the usual pointwise partial order. Also the support functional is affi ne, that is: σ βa+(1 β)b = βσ A + (1 β)σ B. Let C denote the subset of C( N ) that σ maps K( (Z)) into, that is C {σ A C( N ) : A K( (Z))}. Using this mapping and the Weak Order and Continuity axioms, DLR (2001) construct the continuous linear functional W : C R that represents the preference over K( (Z)): and W (σ A ) W (σ B ) if and only if A B. (18) As in the main text, define v : (Z) R to be the restriction of W to the set of support functions of the singleton sets: v(x) W (σ {x} ). It can be shown using the linearity of the support functions that v is affi ne, that is v(βx + (1 β)y) = βv(x) + (1 β)v(y). In addition, as mentioned in ection 2, there exists s N and λ 0 such that v(x) = λ N k=1 xk s k for all x (Z). Dekel, Lipman, Rustichini and arver (2007) show in the proof of their Theorem 2 that under Monotonicity, the functional W is increasing on the space H = {r 1 σ 1 r 2 σ 2 : σ 1, σ 2 C and σ 1, σ 2 0} which is dense in C( N ). ince f f 1 for any f H, where 1 is the function identically equal to 1, by the monotonicity of W we will have W (f) f W (1) so W is bounded on H. Therefore, as in DLR (2001), W can be extended uniquely from C to the whole C( N ) preserving continuity and linearity. Also, since H is dense in C( N ), it follows immediately that W will be monotone on the whole C( N ). As in Royden (1988, page 355), W can be decomposed as W = W + W where W + and W are two positive linear functional forms. Using again the monotonicity of W and the definition of W + from Royden (1988) it is straightforward to show that W ( ) = W + ( ) and W ( ) = 0 on C( N ). Then, W is a positive linear functional on C( N ) so since N is compact, the functions in C( N ) have compact support since closed subsets of compact spaces are compact, so the 13 Note that U(x, s)µ(ds) is a linear function in x so even when A is not convex, the minimum of U(x, s)µ(ds) over A will be attained at an element of A. Thus, the reference point will always belong to A. 19
Riesz-Markov Theorem from Royden (1988, page 352) can be used to write W (f) as an integral of f against a σ-additive positive measure µ over N for any f C( N ). In particular, W (σ A ) = σ A (s)µ(ds) for any A K( (Z)) (19) N This last step delivers the DLR representation of the preference. However, note that we use here a different version of the Riesz Representation Theorem than the one used in DLR (2001). This is because the Monotonicity Axiom makes the functional W positive and thus we can obtain a σ-additive and positive Borel measure as opposed to a finitely additive and signed measure as in DLR (2001). As it will be seen below, the σ-additivity of the measure is necessary both for obtaining our reference-dependent representation as well as for proving the uniqueness of this representation. Next, we will impose the additional restrictions on preferences given by Axiom CEB and Axiom CEB-2 to obtain our specific representation from (6). A2. Proof of Theorem 13 As a first step in the proof, we will rewrite Axiom CEB and Axiom CEB-2 by using the support functionals and the functional W instead of the preference relation. Note that since v(x) represents the preference over lotteries in (Z), using the results from Appendix B1 we have: N N {x} {y} λ y k ( s k ) λ x k ( s k ) λσ {y} ( s ) λσ {x} ( s ) (20) k=1 k=1 Given two sets A, B K( (Z)), if there exists y B such that {x} {y} for all x A we will have that λσ {y} ( s ) > λσ {x} ( s ) for all x A so λσ B ( s ) > λσ A ( s ). Thus, in general if there exists a lottery in B that is strictly worse than all lotteries in A we can write this in a compact way as λσ B ( s ) > λσ A ( s ). imilarly, if y is weakly worse than all elements in A, we have λσ B ( s ) λσ A ( s ). Also note that in order for Axiom CEB-2 to hold, more exactly for a lottery y (Z) to exist such that {x} {y} for some other x (Z), we need λ > 0 since otherwise all elements in (Z) are indifferent to each other. Therefore, under Axiom CEB-2 we have λσ B ( s ) > λσ A ( s ) if and only if σ B ( s ) > σ A ( s ). Finally, for any two support functionals σ A, σ B C, denote their join by σ A σ B, that is (σ A σ B )( ) max(σ A ( ), σ B ( )) and note that σ A B = σ A σ B. Using these results, the fact that A hull(a) for any A K( (Z)) and the fact that A B iff σ A σ B, we can write Axiom CEB and Axiom CEB-2 in the following equivalent forms: 20